8141 2020

March 2020

Gini and Optimal Income Taxation by Rank Laurent Simula, Alain Trannoy

Impressum:

CESifo Working Papers ISSN 2364-1428 (electronic version) Publisher and distributor: Munich Society for the Promotion of Economic Research - CESifo GmbH The international platform of Ludwigs-Maximilians University’s Center for Economic Studies and the ifo Institute Poschingerstr. 5, 81679 Munich, Germany Telephone +49 (0)89 2180-2740, Telefax +49 (0)89 2180-17845, email office@cesifo.de Editor: Clemens Fuest https://www.cesifo.org/en/wp An electronic version of the paper may be downloaded · from the SSRN website: www.SSRN.com · from the RePEc website: www.RePEc.org · from the CESifo website: https://www.cesifo.org/en/wp

CESifo Working Paper No. 8141

Gini and Optimal Income Taxation by Rank

Abstract We solve the non-linear income tax program for a rank-dependent social welfare function à la Yaari, expressing the trade-off between size and inequality using the Gini or related families of positional indices. The key idea is that when agents optimize and absent bunching, ranks in the actual and optimal allocations become an invariant dimension. This allows us to obtain optimal marginal tax rates as a function of ranks, and numerically illustrate the relationship between ranks and taxes. For singles without children, the actual US tax schedule seems to indicate a distaste for differences in the upper part of the distribution.

JEL-Codes: D630, D820, H210.

Keywords: Gini, optimal taxation, income taxation, ranks.

Laurent Simula

ENS de Lyon & UMR GATE University of Lyon / France laurent.simula@ens-lyon.fr

Alain Trannoy CNRS & EHESS

Aix-Marseille University (AMSE) / France alain.trannoy@univ-amu.fr

March 3, 2020 We thank Rolf Aaberge, Nathaniel Hendren, Thomas Piketty, Emmanuel Saez and participants in the Oslo Workshop on ”Inequality: Measurement, Evolution, Mechanisms, and Policies” for helpful comments and discussion. A part of this research was realized while Laurent Simula was visiting the department of economics, Harvard University. We acknowledge financial support from IDEX-University of Lyon within the Programme Investissements d’Avenir (ANR-16-IDEX-0005).

I. INTRODUCTION

”The questions therefore arise what principles should govern an optimum incometax; what such a tax schedule would look like; and what degree of inequality wouldremain once it was established.” (Mirrlees, 1971)

As emphasized by Mirrlees, the trade-off between efficiency and the degree of inequality isat the core of optimal nonlinear income taxation. However, quite surprisingly, the optimal tax-ation and inequality measurement literatures have so far remained largely disconnected. Fromthe very beginning, optimal income tax models have primarily relied on social welfare as mea-sured by a concave transformation of individual utilities. Atkinson (1970) connected inequalityand social welfare by means of the introduction of the equally distributed equivalent level ofincome (see also Blackorby and Donaldson (1978); Blackorby et al. (1999)). However, this the-oretical relationship has hardly been used in practice to infer the level of optimal inequalities(see Sattinger (2017)). Moreover, it sets aside the notion of position, i.e. the ranking of indi-viduals, and more specifically the Gini inequality index and its extensions, to focus on incomelevels. Yet, the Gini index and its graphical representation through the Lorenz curve are by farthe most used tools to measure inequalities.

This article adopts a rank-based approach to the optimal nonlinear income tax problem.Ranks play a key part when thinking about inequalities. Chetty et al. (2014) have recentlyshown how ranks help obtain clear-cut results with regards to intergenerational mobility. Inthe same perspective, we show that a rank-dependent social welfare function a la Yaari (1987,1988) allows us to obtain very simple tax rate formulas, and empirical results clearly expressingthe trade-off between the ”size of the cake” and its uneven sharing, as measured by the Ginicoefficient of utility levels, or its extensions to the S-Gini family (Donaldson and Weymark,1980; Weymark, 1981; Bossert, 1990) and ”A-family” (Aaberge, 2000, 2009; Aaberge et al., 2020).

We exploit the key feature that in a social optimum respecting incentive-compatibility con-straints, ranks in terms of skills, indirect utilities, gross incomes and net incomes are all thesame, as emphasized by Trannoy (2019). Furthermore, absent bunching, a mild assumption inview of empirical findings (Saez, 2010; Bastani and Selin, 2014), these ranks are the same asin the actual allocation providing agents maximize their utility. Ranks thus provide a fruitfulinvariant with respect to which tax rates can be expressed. Therefore, far to being farfetched,rank-dependent social welfare functions appear to be a relevant criterion to assess optimality ofan incentive-compatible income tax. Social welfare is then equal to a weighted sum of indirectutilities –with weights solely depending on ranks– or equivalently using its ”abbreviate” formas the mean indirect utility multiplied by one minus the corresponding inequality index.

Relying on the Gini social welfare function introduced by Sen (1974), we first investigate thesituation in which the inequality index is the popular Gini coefficient.1 In that case, individualweights in the social objective vary linearly with the rank. We also consider extensions to theS-Gini and A-families, two single-parameter families sharing the Gini social welfare functionas unique common element. In the former, individual weights are convex with respect to therank, leading to downward positional sensitivity; they are concave in the latter, leading to up-ward positional sensitivity (Aaberge, 2009; Aaberge et al., 2020). This implies that Pigou-Daltontransfers should be prioritized when taking place at the bottom for the S-Gini family, and at the

1Our approach remains welfarist. See Prete et al. (2016) for an analysis using the Gini index in a non-welfarist setting.

2

top for the A-family.Following Piketty (1997), Diamond (1998) and Saez and Stantcheva (2016), we assume away

income effects on taxable income. We show that optimal marginal tax rates can be expressedin a very simple way as functions of ranks. On this basis, we establish that marginal tax pro-files depend on ranks in a decreasing or U-shaped manner, for all members of the S-Gini andA-families. We even obtain a closed-form expression for Gini tax liabilities, providing the elas-ticity of taxable income is constant and ranks follow a Pareto distribution. We then numericallyillustrate the relationship between ranks and taxes. We parameterize the formulas based onUS data for singles without children and an elasticity of taxable income in the 0.1-0.5 range.Although the Gini social objective function appears on paper as a natural intermediate case be-tween the Rawlsian maximin and pure utilitarianism, we find that the Gini optimal tax scheduleis much closer to the former. Moreover, the schedules obtained for the different members of theS-Gini class are rather close. On the contrary, the A-family offers much more dispersed viewsregarding how marginal and average tax rates should vary with ranks. It seems that the rela-tive concern for top people matters more for the design of the optimal non-linear income taxschedule than the concern for bottom people. Furthermore, we show –admittedly for a specificpopulation– that the US very unlikely optimizes with the Gini coefficient in mind, but with amember of the A-family, which expresses a particular concern for inequalities at the top. Still,the approach delivers optimal marginal tax rates about ten points higher than the actual onesfor top-income earners for a constant elasticity of taxable-income equal to 0.5 and, therefore, asignificantly more progressive tax schedule at the top. This calls for complementary and morecomprehensive simulation exercises, for the US but also for other developed countries.

Our approach illustrates the fruitfulness of the ”generalized social weights approach” toincome taxation introduced by Saez and Stantcheva (2016), on which we build extensively, eventhough the latter article only introduced ranks in relation to equality of opportunity principles.

The article is organized as follows. Section 2 recalls the concepts of rank-dependent socialwelfare functions and their connection with inequality indexes. On this basis, Section 3 formu-lates the optimal income tax problem in terms of ranks. Section 4 develops the main theoreticalresults. Section 5 empirically examines the link between ranks and optimal taxes. Section 6concludes.

II. RANK-DEPENDENT SOCIAL WELFARE ANDINEQUALITY

This section introduces rank-dependent social welfare functions and shows how they connectwith inequality indexes.

II.1. Rank-Dependent Social Welfare

We consider a population of individuals, heterogeneous with respect to a variable x. For sim-plicity, we assume that the latter is uni-dimensional and smoothly distributed according to thecumulative distribution function F(x), with support X ⊆ R+. We call f (x) the correspondingprobability density function (pdf). This continuous setting is a good approximation of a largediscrete population, makes the analysis more straightforward, and will later allow us to make

3

the connection with marginal tax rates, which are not well-defined in a discrete setting (cf. Wey-mark (1987) or Simula (2010)). The average value of x within the population is µ =

∫X x f (x)dx.

We define the quantile function as F−1(p) = x, where p ∈ [0, 1] stands for the rank or”position”, and introduce weights to capture the social planner’s aversion to inequality. Themarginal weights are denoted λ(p) and the cumulated weights Λ(p) =

∫ p0 λ(π)dπ. In the

whole article, we focus on weights consistent with second-order stochastic dominance, belong-ing to the set:

L = {∀p ∈ (0, 1), λ(p) > 0 and λ′(p) < 0; Λ(0) = λ(1) = 0; Λ(1) = 1}. (1)

The assumption that λ(p) is positive and decreasing means that every individual counts, butto a lower extent the higher the rank. The other assumptions are normalizations. On this basis,rank-dependent social welfare (Yaari, 1987, 1988) is defined as:

W =∫ 1

0λ(p)F−1(p)dp. (2)

II.2. From Rank-Dependent Social Welfare to Inequality Indexes

The distribution F is egalitarian if and only ifW = µ. Otherwise, there is a positive gap betweenwelfare W and the equality benchmark, denoted ∆ ≡ µ −W . Dividing by µ, we obtain themean-invariant inequality index I ≡ ∆/µ. Combining the definitions of ∆ and I, we can rewritesocial welfare (2) in abbreviated form:

W = µ(1− I). (3)

This expression illustrates the close connection between rank-dependent social welfare func-tions and inequality indexes: W is equal to the egalitarian benchmark deflated by inequality asmeasured by I.

Specifying the weights Λ(p), we focus on two important families of rank-dependent socialwelfare functions:

- The S-Gini family (Donaldson and Weymark, 1980) for Λ(p) = 1− (1− p)δ and δ ≥ 2.

- The ”A” family introduced by Aaberge (2000)2 for Λ(p) = (δp− pδ)/(δ− 1) and δ ≥ 2.When δ → 1, Λ(p) = p(1− log(p)) which corrresponds to the rank-dependent welfarefunctionW in which I is the Bonferroni index of inequality.3

For both families, the weights coincide when δ = 2, with Λ(p) = p(2− p). In that case, theinequality measure I is the Gini coefficient andW the Gini social welfare function introducedby Sen (1974). The marginal weights λ(p) = 2(1− p) are then linear with respect to rank. Bycontrast, for any δ > 2, the weights λ(p) are convex when considering the S-Gini family, andconcave for the A family. Figure I shows the weights for both families, the common Gini case,and the Bonferroni case. The left panel shows individual weights λ(p). Because they all belong

2Aaberge (2000) refers to this class as the Lorenz family for δ = 2, 3, .... It corresponds to the integer subfamily ofthe ”illfare-ranked single-series Ginis” discussed by Donaldson and Weymark (1980) and Bossert (1990). We refer to thelatter as the ”A” family. Aaberge (2009) and Aaberge et al. (2020) have indeed shown the usefulness of this family foranalyzing inequality when Lorenz curves intersect.

3Let µ(x) =(∫ x

0 xdF(x))

/F(x) and r(x) = (µ− µ(x))/µ. The Bonferroni inequality index is∫

X r(x)dF(x).

4

S-Gini for 𝛿 = 3

A-family for 𝛿 = 3

Gini

Bonferroni

0.2 0.4 0.6 0.8 1.0p

0.2

0.4

0.6

0.8

1.0

�(p)

A, 𝛿=10

S-Gini, 𝛿 = 10

Rawls

GiniBonferroni

Utilitarianism

p

Λ(𝑝)

0.2 0.4 0.6 0.8 1.0p

1

2

3

�(p)

p

λ(𝑝)

FIGURE I: WEIGHTS TO RANKS

to L, the area below each curve is equal to one. Looking at the different curves, let us considera fixed transfer taking place between two agents with equal difference in ranks. For the S-Ginifamily, convexity implies that the equalizing effect of the transfer becomes larger the lower theranks considered. The focus is thus on poverty. On the contrary, for the A family, the higherthe ranks the stronger the equalizing effect. The emphasis is layed on inequalities at the topof the distribution. The right panel shows cumulated weights Λ(p) for various values of δ. Itillustrates the following convergence pattern when δ goes up: the S-Gini and A-families tendto two important benchmarks, the Rawlsian maximin on the one hand and pure utilitarianismon the other hand. It also shows that, for every interior p, the cumulated weights are larger inthe Bonferroni case than in the Gini one, implying that the former stresses poverty more thanthe latter.

III. OPTIMAL NON-LINEAR INCOME TAXATION ANDINEQUALITY MEASUREMENT

We now introduce taxes and transfers, and thus the main elements needed to investigate thetrade-off between equity and efficiency, in the tradition of Mirrlees (1971) and Diamond (1998).We consider a population with a continuum of individuals and size normalized to one. Thispopulation is heterogeneous with respect to productivity per unit of effort, θ, which belongsto a subset

[θ, θ]

of R+. θ can be finite or tend to +∞. Productivity is privately known toeach agent, and only its cumulated distribution H(θ) (with pdf h(θ) = H′(θ) > 0) is commonknowledge. Every individual derives utility from consumption c and dis-utility from earningincome z, with a utility representation:

u(c, z; θ) = c− v(z; θ). (4)

5

The function v is common to all individuals. It is increasing and convex in z, decreasing inθ, and with a negative cross-derivative. The quasi-linear form (4) rules out income effects onearnings, and greatly simplifies the analysis by allowing us to focus on the distortive impact ofsubstitution effects. The government sets an income tax T(z) as a function of earnings only, sothat c = z− T(z). Individuals choose z to maximize u = z− T(z)− v(z; θ). We call z(θ) thesolution and V(θ) the corresponding indirect utility.

Because of asymmetric information, the government must account for incentive-compatibilityconstraints when designing the income tax T(z). As is well-known (see, e.g., Salanie (2011)),these constraints are equivalent to (i) V′ (θ) = −v′θ (z (θ) ; θ) and (ii) non-decreasing z(θ) (equiv-alent to non-decreasing c(θ) = z(θ)− T(z(θ))). The first condition implies that the higher theproductivity the larger the indirect utility. Consequently, the ranks in terms of productivity andthe ranks in terms of indirect utility are the same. Absent bunching, the latter are also the sameas for gross income and net income. In the theoretical derivations, we assume that there is nobunching, and check it ex post in the numerical simulations of Section V.

In reference to the previous Section, we let x = V. Because of the above remark about ranks,we note that p = F(V(θ)) = H(θ), which also implies: F−1(p) = V(θ). Consequently, therank-dependent social welfare function (2) can be rewritten as follows:

W =∫ θ

θλ(F(V(θ)))V(θ)dF(V(θ)) =

∫ θ

θλ(H(θ))V(θ)dH(θ). (5)

The government’s problem is to design the tax function T(z) maximizing (5) subject to the con-ditions for incentive compatibility and budget-balancedness. We call E the exogenous amountof expenditures to finance if any. In this form, we see the clear connection with the probleminvestigated by Saez and Stantcheva (2016), with ”generalized welfare weights” which in thepresent article only depend on productivity. What is novel is that we offer a new perspective,connecting optimal taxation and inequality measurement thanks to the rank-dependent formu-lation of the social objective. Such an analysis was not previously made in the literature.

Remarks

The rank-dependent social welfare functions we consider depend on a single parameter δ. Onemay therefore wonder whether the solutions to the above optimal income tax problem may notbe obtained as the solution to the maximization of Atkinson social welfare function,

WAtk =∫ θ

θ

11− ρ

V(θ)1−ρdH(θ),

for a given value of the society’s aversion to inequality ρ. This would be the case if an allocationwas satisfying the optimality conditions for the optimal income tax problem under theW andWAtk objective functions. For that to be the case, Euler equations should be the same, imply-ing λ(H(θ)) = V(θ)−ρ and thus V(θ) = [λ(p)]−1/ρ. Therefore, to get a similar solution, theindirect utility path V(θ) should be completely exogenous, independent in particular of indi-vidual preferences and elasticities of taxable income, which is impossible. This establishes thespecificity of the rank-dependent approach within the class of welfare functions depending ona one-dimensional parameter.

6

In addition, the rank-dependent social welfare functions we consider allow a transparentformulation of the optimal income tax problem, because social welfare is simply obtained asthe product of µ and 1− I, as already noted in (2). This is not the case if we instead useWAtk.The closest formulation would involve the equally distributed equivalent income EDEI andAtkinson inequality index IAtk = 1− EDEI/µ. On this basis,WAtk can be rewritten as:

WAtk =1

1− ρ[µ(1− IAtk)]

1−ρ ,

which however is a nonlinear transformation of the product of µ and 1− IAtk.

IV. OPTIMAL MARGINAL TAX RATES BY RANK

Optimal tax formulas are usually presented as a function of earnings in the optimum (see, e.g.,Saez (2001); Piketty and Saez (2013); Saez and Stantcheva (2016)) which are endogenous to theoptimal schedule itself. We derive formulas in terms of rank, a dimension which is completelyexogenous to the optimal tax policy as long as the actual allocation is incentive compatible.

General Case

At a given rank p, we denote the marginal tax rate by MTR(p) and define the elasticity oftaxable income with respect to the retention rate 1−MTR(p) as:

ε(p) =1−MTR(p))

z(p)∂z(p)

∂(1−MTR(p)). (6)

We also introduce the elasticity of taxable income with respect to productivity θ:

εθ(p) =θ(p)z(p)

∂z(p)∂θ(p)

, (7)

with θ(p) = H−1(p). Behavioral responses are summarized by the ratio e(p) = ε(p)/εθ(p),referred to below as the ”behavioral term”.4 When the disutility of effort is isoelastic, given byv(z; θ) = (z/θ)1+1/ε, the behavioral term e(p) and the ”usual” elasticity of taxable income ε(p)are both constant, equal to ε.

In addition, it is useful to introduce the (relative) average social marginal weight for indi-viduals with rank above p,

G(p) =

∫ 1p λ(π)dπ∫ 1

p dπ=

1−Λ(p)1− p

,

and the local Pareto parameter of the skill distribution at rank p,

α(p) =θ(p)h(θ(p))

1− p.

4Our derivation of optimal marginal tax rates is done for the classic case where there are only supply-side responsesas in Mirrlees (1971) and Saez (2001). It is straighforward to extend the analysis to capture other responses, such as taxavoidance and bargaining, as in Piketty et al. (2014)

7

Absent bunching, the optimal marginal tax rates are given by Proposition 2 in Saez and Stantcheva(2016), which remains valid for rank-dependent welfare functions:

MTR(p) =1− G(p)

1− G(p) + e(p)α(p)for 0 < p < 1. (8)

Alternatively, we can rearrange (8) into an ABC-formula.

PROPOSITION 1. For the rank-dependent social welfare functionW , optimal marginal tax ratessatisfy:

MTR(p)1−MTR(p)

=1

e(p)︸︷︷︸A

1α(p)︸ ︷︷ ︸

B

Λ(p)− p1− p︸ ︷︷ ︸

C

for 0 < p < 1. (9)

This formula is extremely simple and straightforward to implement. Compared to Dia-mond’s (1998) formula, the A and B factors are just the same but rewritten in terms of ranks:they reflect efficiency and demographic considerations respectively (see Piketty (1997) or Pikettyand Saez (2013)).5 By contrast, the C-factor summarizing ethical considerations directly de-pends on ranks and on the parameter δ only; and in particular is independent of the marginalutility of income. Under pure utilitarianism, it is equal to zero, so that MTR(p) = 0 for everyrank p. The Rawlsian maximin, often used as a benchmark in the literature (Piketty, 1997; Boad-way and Jacquet, 2008), is easily obtained as a sub-case of our rank dependent setting. The onlyrank which then matters is the lowest; hence, λ(p) = 0 but at θ, implying G(p) = 0 in Formula(8) and C = 1 in Formula (9).

Providing the actual tax schedule is incentive-compatible and absent bunching, the distri-butions of ranks in terms of productivity, gross earning, net earning and indirect utility are allthe same than those in the optimal allocation. Consequently, the B- and C-factors are exoge-nous to the optimal policy. The only factor which may differ is A, which captures efficiencyand therefore evaluates behavioral responses along the optimal schedule. If these responses arestructural, A is also exogenous, and the optimal tax in our rank-dependent setting depends onthe primitives of the model and not on any endogenous variable.

only depends on the actual distribution of p.Figure II shows values of the C-factor by rank p, for various situations: S-Gini and A families

with δ = {2, 3, 9, 15, 21} together with the Bonferroni, Rawlsian and utilitarian cases. It is usefulto recall that MTR increases with the ethical C-factor, everything else being equal, and that thehighest incentive-compatible marginal tax rates are obtained for the Rawlsian maximin. Forthe Gini social welfare function, C is simply equal to the rank p. The contrasting shapes ofC echo our above remark on the main relative focuses on poverty and inequalities at the top,for the S-Gini and A families respectively: in the S-Gini case, C tends to increase the marginaltax burden in the left part of the distribution to increase collected taxes and thus the transferto the poorest; in the A-family case, most of the increase takes place for large ranks. When δ

becomes arbitrarily large and for all interior p, the C-factor converges to its Rawlsian value forthe S-Gini family and to its utilitarian value for the A-family.6 The Bonferroni case appears as

5The A factor in Diamond’s (1998) formula is written in terms of elasticity of labor supply. Proposition (9) uses theequivalent formulation in terms of elasticity of taxable income.

6When δ → ∞, MTR(p) is zero for every p < 1 and then equal to the Rawlsian one at the limit, when p → 1. In thelimit, the graph of the C-Factor is thus inverted-L shaped.

8

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0 p0

C-Factor

A-Family

S-GiniFamily

Gini

Bonferroni

δ =3

δ =9

δ =15

δ =21

𝛿 =3

𝛿=9

𝛿=15

𝛿=21

Rawls

Utilitarianism

FIGURE II: C-FACTOR IN ABC TAX FORMULA (FOR INTERIOR RANKS)

an intermediate situation; if we compare it to S-Gini with δ = 3, the C-factor is larger for pbetween 0 and p ≈ 0.32, and lower for larger values of p.

Heuristic Derivation

It is insightful to rely on a small tax reform perturbation to contrast Formula (8) with thatobtained in the standard ”welfarist” approach. To this aim, let us consider a given tax scheduleand increase marginal tax rates by an amount ∆ on a small interval [p, p + dp]. This gives riseto the following effects:

- Behavioral effect: an agent with rank p in the interval responds to the rise in the marginaltax rate by a substitution effect. Given the definition of ε, the latter reduces her taxableincome by:

dz(p) =z(p)

1−MTR(p)ε(p)∆.

This decreases the taxes she pays by an amount:

dT(p) = MTR(p) · dz(p) =MTR(p)

1−MTR(p)z(p)ε(p)∆.

The mass of taxpayers in the interval is dp = h(θ)dθ. Hence, using (7),

dθ(p) =θ(p)z(p)

dz(p)εθ(p)

,

the total behavioral response amounts to dp · dT(p), i.e.,

dB =MTR(p)

1−MTR(p)ε(p)εθ(p)

h(θ(p))θ(p)dz(p)∆.

9

- Mechanical effect: every agent with rank above p faces a lump-sum increase in taxes,equal to dz(p)∆. Collected taxes increase by

dM = dz(p)[1− p]∆.

- Welfare effect: the lump-sum increase in tax liabilities faced by agents with rank above pinduces a negative income effect, which reduces their indirect utility V(θ(p)) by dz(p)∆.Social welfare thus varies by:

dW = −∫ 1

pλ(p)h(θ(p))∆dz(p).

A small tax reform perturbation around the optimal schedule has no first-order effect. There-fore, optimal marginal tax rates must verify dB+ dM + dW = 0, and thus the formula in Propo-sition 1.

The welfare effect dW is negative as in the standard welfarist framework where the socialobjective function is the sum of a concave transformation Φ(V(θ(p))) of individual utilities.In the latter, a lump-sum increase in tax liabilities at skill θ(p) reduces indirect utility V(θ(p)),thus increasing the social marginal utility Φ′(V(θ(p))). This ”feedback” effect does not arise inthe rank-dependent approach: because the lump-sum increase in tax liabilities has no effect onranks p, there is no variation in λ(p).

Tax Rates at the Top and at the Bottom

To investigate tax rates at the top, we consider a constant behavioral term e and a rank p abovewhich H is well-approximated by a Pareto distribution (Saez, 2001).7 The latter implies thatα(p) is constant above p, equal to α. We consider Bonferroni as well as the S-Gini and A-families for finite values of δ. In all cases, the increase in the C-factor emphasized above implieshigher marginal tax rates for larger ranks. We therefore generalize the results of Diamond(1998) over increasing marginal tax rates at the top to a broad class of alternative social welfarefunctions. Moreover, we are able to rank the marginal tax rates above p on the basis of FigureII: for example, the marginal tax rates are always larger for the S-Gini family of social welfarefunctions than for the A-one; also, at a given rank, the marginal tax rate increases with δ forthe S-Gini family and decreases with it for the A-family. When p → 1, they all converge to theRawlsian one, where MTR = 1/(1 + e · α).

We now turn our focus to the bottom of the distribution, assuming again that the behavioralterm e(p) is constant. The ranking of the C-Factors as shown in Figure II is preserved when ptends to 0. Indexing the marginal tax rates by their family, we obtain:

1 = MTRR(0) > MTRSG(0) > MTRG(0) > MTRA(0) > 0.

In addition, the higher δ in the S-Gini family, the closer MTRSG(0) to 1; and the higher δ in theA-family, the closer MTRA(0) to 0. Therefore, while marginal tax rates for all families tend toconverge at the top (for a Pareto distribution), there is a grading of marginal tax rates at thebottom.

7The Pareto distribution is unbounded with θ → ∞.

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The Gini Benchmark

The optimal tax formulas can be written in a strikingly simple form for the Gini social welfarefunction. In that case, G(p) = 1− p. Therefore, applying Formulas (8) and (9), we obtain:

PROPOSITION 2. For the Gini social welfare function, optimal marginal tax rates satisfy:

MTR(p) =p

p + e(p)α(p)and

MTR(p)1−MTR(p)

=p

e(p)α(p)for 0 < p < 1. (10)

When behavioral responses are structural (in the sense that p → e(p) is fixed at every p),the ratio of the marginal tax rate to the retention tax rate for the Gini social welfare function isequal to that of the Rawlsian maximin, multiplied by p. As a result, for rank intervals wherethe Rawlsian tax schedule is marginally progressive (i.e., MTR(p) increasing in p), so is theGini schedule. If we further assume that e(p) and α(p) are constant above productivity θ( p),respectively equal to e and α, the tax liability T(p) is obtained by integration over ranks, whichby essence are uniformly distributed over [0, 1]. Hence,

T(p) = p− eα log(p + eα) + T,

where T is a constant. If E is the tax receipts to be levied from agents above p, we obtain

T = eα [log(1 + eα))− 1] + (eα)2 [log(1 + eα)− log(eα))] + E− 12

.

If the whole distribution of productivity is Pareto and the tax policy purely redistributive (E =

E = 0), we obtain a full characterization of the tax function T(p), with increasing marginal taxrates.

V. NUMERICAL ILLUSTRATION

We numerically illustrate the link between ranks p and optimal taxes. We consider that thedisutility of effort is isoelastic, with v(z; θ) = (z/θ)(1+1/ε). In that case, the elasticity of taxableincome ε(p) is constant, equal to e(p). The calibration is the same as in Lehmann et al. (2014)for a closed economy, with the distribution of productivity obtained by inversion from the CPSdata (2007) for singles without children, extended by a Pareto tail. In that sense, we ”replicate”a US economy populated with this specific set of agents.

The elasticity ε is assumed to be in the range 0.25− 0.5 (Saez et al., 2012). Regarding theamount of exogenous expenditures to be financed (E), we consider two scenarios: it is equalto zero in the first one, where the tax policy is purely redistributive; it is equal to the averageof the tax liabilities actually paid by the agents in our sample in the second one (E = Tactual).The chosen scenario is without impact on the marginal tax rates, but affects average tax ratesand therefore net earnings and indirect utilities. The first situation puts the emphasis on thenormative dimension of the simulation exercise and is typically used as a benchmark in theoptimal taxation literature. The second one allows us to more directly relate the ”optimal” and”actual” schedules, given that little vertical redistribution actually takes place within singleswithout children, as illustrated by the shape of the observed average tax rates in Figure IV.

11

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MTR

Rawls

Bonferroni

S-Gini, 𝛿 = 3

Gini

A-family, 𝛿 = 3

A-family, 𝛿 = 16

≈55.8

p

MTR

Rawls Bonferroni

S-Gini, 𝛿 = 3

Gini

A-family, 𝛿 = 3

A-family, 𝛿 = 16

≈71.6

p

e= 0.25 e= 0.5

Observed MTR

FIGURE III: OPTIMAL MARGINAL TAX RATES (IN %) BY RANK

Based on this numerical exercise, we hope to generate interest for ”alternative” welfare func-tions which may be closer to the social planners’ actual preferences, as emphasized below, andpave the way to more precise and complete simulations, with better calibrations, which how-ever are beyond the scope of the present insight paper.

Figure III shows optimal marginal tax rates for the the different families of rank-dependentsocial welfare functions considered above. The left panel is obtained for an elasticity e = 0.25;the right panel for e = 0.5. We also report the ”observed” marginal tax rates. On both panelsand for the S-Gini, Gini and Bonferroni cases, the optimal marginal tax rates are decreasingfor about 80% of the ranks, and then slightly increasing, converging to the Rawlsian marginaltax rate at the very top. When δ increases, the marginal tax rates for the S-Gini family rapidlybecome very close to those obtained under the Rawlsian maximin. The pattern looks verydifferent for the A-family. The marginal tax rates are really U-shaped, with a marked declineat the bottom, getting steeper the larger δ, and then a sharp increase for ranks above .8. Forexample, when δ = 16 and e = 0.25, the marginal tax rates are multiplied by about two in thetop 20% interval. On both panels, we see that the A-family generates the optimal tax profileswhich are the closest to the observed one. When e = 0.5 and δ = 16, the optimal marginal taxrates are increasing for more than 90% of the ranks, with an initial decline that would probablybe phased out if participation margins were accounted for; for the top 10% however, the optimaltax rates become significantly larger than the observed ones, with a difference going up to 11percent points.

Figure IV shows the optimal and observed average tax rates when e = 0.25, for E = 0 inthe left panel and E = Tactual in the right panel. Let us first focus on the Rawlsian, S-Gini andBonferroni families. The average tax rates are close at ranks where they are positive. Whenthe tax policy is purely redistributive as in the left panel, the average tax rates are negative forthe bottom 75% and sharply increasing thereafter, reaching 71.3% at the top. When exogenous

12

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ATR

p

Bonferroni

Rawls

S-Gini, 𝛿 = 3Gini

A-family

𝛿=3

𝛿=16

≈71.3

≈44

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𝛿 = 16

Rawls

A-family

𝛿=3 S-Gini, 𝛿 = 3

Bonferroni

≈71.5

≈44

p

observedATR

ATR𝐸 = 0 𝐸 = )𝑇+,-.+/

FIGURE IV: AVERAGE TAX RATES IN % BY RANK FOR e = 0.25

taxes have to be levied as in the right panel, the ”break even” point is shifted to the left, withpositive tax rates for about 56% of the ranks. The differences are larger at ranks for which theaverage tax rate is negative. We now consider the A-family, which spans a much broader partof the space, compared to the other families. When δ = 16 and E = Tactual , the optimal andactual schedules are extremely close for 75% of the ranks. As emphasized above, differencesin the bottom 10% would probably disappear if participation margins were accounted for. Themain discrepancy is therefore at the top, especially in the last decile, with observed rates up to27.5 percent points lower than the optimal levels.

As emphasized above, Mirrlees (1971) wondered ”what degree of inequality would remainonce [an optimal income tax] was established”. We provide a graphical answer, assumingE = 0. We want to illustrate the trade-off between equity and efficiency for the various con-ceptions of equality considered above. The value of the inequality index I in W as shown inEquation (3) will vary for each of these conceptions. To make a graphical representation, wemust therefore choose one particular index as a common metric. We choose the popular Giniindex. This is why Figure V shows equity measured as one minus the Gini coefficient of indirectutility, as a function of efficiency measured as the average indirect utility. We show two ”fron-tiers”, corresponding to an elasticity e of 0.25 and 0.5 respectively. Each of them starts with theRawlsian maximin on the upper left and ends up with the laissez-faire/pure utilitarianism onthe lower right.8 The first part of each frontier corresponds to the S-Gini family (in green) andthe second part to the A-family (in purple). When indicated, the numerical labels correspondto the values of δ. We note that the two frontiers are concave, and become roughly linear in theA-family part for large values of δ. This implies ”marginal decreasing returns”, especially inthe left part, where aggregate resources have to be largely reduced in order to set slight gains inequity. We also see that the A-family spans much more diverse situations in terms of inequalitythan the S-Gini family. When e = 0.5, one minus the Gini coefficient of indirect utility varies byabout .03 between the Rawls and Gini situations; and by 0.28 between Gini and Utilitarianism.

8Given that individuals have quasilinear preferences, the laissez-faire and pure utilitarianism coincide.

13

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AverageIndirect Utility

1 – Gini(Indirect Utility)

R 3G

LF

36

9162040

R 3

B

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9162040

LF

e=0.25 and E = 0

e=0. 5 and E = 0

Legend:RawlsS-Gini familyBonferroniGiniA-familyLaissez-Faire

FIGURE V: EQUITY VS EFFICIENCY TRADE-OFF (E = 0)

Alternatively, we could show very similar graphs with one minus the Gini coefficient of netincome as a function of average gross income (cf. Appendix). When E = Tactual , e = 0.25 andWis the Gini social welfare function, the Gini coefficient of net income amounts to 0.252 (against0.481 in the actual distribution) and the mean gross income to $42, 916 (against $51, 410).

VI. CONCLUSION

This article can be viewed as going one step further in the list of studies which, starting fromMirrlees (1971), have attempted to get more readable results for optimal non-linear incometaxes. The main milestones on this road were Piketty (1997), Diamond (1998), Saez (2001) andSaez and Stantcheva (2016). Our contribution is to capture the ethical advantages of redistribut-ing across ranks, a dimension which is completely exogenous to the optimal tax policy as longas agents optimize and there is no bunching. This allows us to express optimal tax rates in anextremely simple way.

The Rawlsian maximin is often used as a benchmark in the optimal taxation literature. Weprovide alternative benchmarks, based on ranks, and especially the Gini welfare function em-phasized by Sen (1974), in which the average multiplied by one minus the Gini coefficient ismaximized. We show that the S-Gini family spans the space between Gini and the Rawlsianmaximin, while the A-family spans the gap between Gini and pure utilitarianism.

Optimal income taxation has often been criticized on the ground that it generates scheduleswith little resemblance to the actual ones. Our numerical simulations show that social welfarefunctions from the A-family are more likely to generate marginally increasing tax schedulesfor the entire population, as typically observed in reality (when focusing on the ”payment”

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side of the tax-and-transfer system). Moreover, for a specific population, we show that theUS very unlikely optimizes with the Gini coefficient in mind, but rather with a member ofthe A-family, laying emphasis on differences occurring in the upper part of the distribution;however, top tax rates appear to be significantly larger than in the actual schedule. This callsfor complementary and more comprehensive simulation exercises, for the US but also for otherdeveloped countries.

Appendix (Meant to be Online)

Figure VI shows the trade-off between equity and efficiency with respect to two observabledimensions, gross income and net income.

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Average Gross Income ($/year)

e=.25, E=0

e=.5, E=0

Legend:RawlsS-Gini familyBonferroniGiniA-familyLaissez-Faire

3

3

3

9

9

9

3

3e=.25, 𝐸 = %𝑇'()*'+

9

3

16

16

16

e=.1, E=0

16

1 – Gini(Net Income)

3

3

FIGURE VI: EQUITY VS EFFICIENCY TRADE-OFF

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